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When Sannecherib, King of Assyria, arose, he confused the identity of all the nations, mixing them together, and exiling them from their place. The Egyptians that live in the land of Egypt at present are of other nationalities. This also applies with regard to the Edomites in the field of Edom.

Since these four forbidden nations became intermingled with all the nations of the world [with] whom it is permitted [to marry once they convert], all [converts] are permitted. For when anyone of them separates himself [from them by] converting, we operate under the presumption that he became separate from the majority. Therefore in the present age, in all places, whenever a convert converts, whether he be an Edomite, an Egyptian, an Ammonite, a Moabite, a Kushite, or from any of the other nations, whether male or female, he or she is permitted to marry among the Jewish people immediately.

Translation note: Rambam didn't say "discriminated-against peoples", he said "prohibited peoples." Also the last sentence should read "whether Edomite, Egyptian, Ammonite, Moabite, Kushite, or any other people" (I assume the Rambam is saying not to guess true ancestry by race, which is scientifically correct).
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ShalomJul 8 '11 at 14:31

Clarify: "the majority of the world's non-Jewish population is not prohibited [after converting], so now that the minority [Egypt Edom Amon Moab] mixed into them, we assume any individual comes from the majority population and is permitted."
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ShalomJul 8 '11 at 14:33

I'm really not sure what DNA has to do with this. One question is using DNA evidence as proof of death; another is questions of paternity (but even if Joe Jew finds his paternal DNA doesn't match his "dad" he's still not a mamzer, as maybe it was a non-Jew, they doesn't generate mamzerut). If someone shows up today and claims to be an Edomite, you'd have to show that his father's father's father's father ... (it's patrilineal) was a pre-Sennacherib Edomite. That's only feasible in sci-fi, to the very best of my knowledge.
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ShalomJul 8 '11 at 14:40

Nobody is currently willing to declare a person a cohen or not a cohen based on their dna. In theory, you could find the graves of edomites, look at the DNA and find ways of discovering who is a descendant and who is not based on the Y chromosone (which only comes from the father). But nobody wants to do that.
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aviJul 8 '11 at 14:46

Perhaps I should ask this as a separate question, but it directly affects the value of this answer to the present question, so I'll ask it here instead: What does the Rambam do with the g'mara in M'gila that says that Moav (see Tosafos) was not moved around by Ashur (or anyone else who came along, at least until the Purim story occurred)?
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msh210♦Jul 8 '11 at 15:11